Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

J. Pestana

doi:10.1137/18M1205406

Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

J. Pestana

Mathematics And Statistics

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

52 Downloads (Pure)

Abstract

When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.

Original language	English
Pages (from-to)	870-887
Number of pages	18
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	40
Issue number	3
DOIs	https://doi.org/10.1137/18M1205406
Publication status	Published - 16 Jul 2019

Keywords

Toeplitz matrix
multilevel Toeplitz matrix
symmetrization
preconditioning
Krylov subspace method

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Pestana-SIAM-JMAA-2019-Preconditioners-for-symmetrized-Toeplitz-and-multilevel-ToeplitzFinal published version, 533 KBLicence: CC BY 4.0

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title = "Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices",

abstract = "When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments. ",

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AU - Pestana, J.

PY - 2019/7/16

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N2 - When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.

AB - When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.

KW - Toeplitz matrix

KW - multilevel Toeplitz matrix

KW - symmetrization

KW - preconditioning

KW - Krylov subspace method

UR - https://www.siam.org/Publications/Journals/SIAM-Journal-on-Matrix-Analysis-and-Applications-SIMAX

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DO - 10.1137/18M1205406

M3 - Article

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VL - 40

SP - 870

EP - 887

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -

Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

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Jennifer Pestana

Effective preconditioners for linear systems in fractional diffusion

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Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

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Jennifer Pestana

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Effective preconditioners for linear systems in fractional diffusion

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